Ura-zuki is more linear. Age Ura-zuki is rising but still linear. What about Sukui ura-zuki? Any martial artists know this one? I am hoping for an official karate term, but I will settle for a gramatically correct term that describes a scooping punch (western boxing uppercut). Thank you in advance!!
I don't do karate or any other martial arts, but the basic term for this is katakana アッパーカット, which should be understood by almost everyone who is fluent in Japanese.
裏拳 (ura-ken) is more a like backhand blow, or a blow using the back of one's hand. This is well-known, too.
裏突き (ura-zuki) is a straight punch with the palm facing upward. I think it's a karate-specific term.
According to this list of karate 突き, it appears 鉤【かぎ】突【づ】き (kagi-zuki, 鉤 means 'hook') clearly says it's a 弧を描くような (=curved, arc-like) punch, but it's a punch blown horizontally. It looks like a karate jargon rather than a generic term, but I don't know the awareness of this term even among karate fans.
I haven't heard age ura-zuki (上げ裏突き?) or sukui ura-zuki (すくい裏突き?) and they're not listed in the linked list, either (these names look reasonable at least to me, though).